Vibrational probability function

Figure 2.4 Vibrational probability functions for a series of vibrational quantum numbers. Note that for the higher v there is a greater probability of the molecule having a bond length at the two limits shown by the Morse curve. Notice also that for each value of v, there are v + 1 maxima...

Understand the importance of the overlap of vibrational probability functions and the energy gap law in determining the rate of internal conversion and intersystem crossing. 

To illustrate the first point concerning a spectator bond for the abstraction reaction, Fig. 17 shows the total reaction probability for the abstraction reaction as a function of the translational energy for total angular momentum J = 0 on the YZCL2 PES with the H20 reactant in the ground rovibrational state [the (00)(0) state in the local mode notation], where the uncleaved bond OHb is treated in various ways. Using a limited number of one or five vibrational basis functions, VBF(OHb) = 1 or 5, means that the OHb bond is unreactive, a spectator. The abstraction reaction probability... 

The modulation of the charge of the adsorbed atom by the vibrations of heavy particles leads to a number of additional effects. In particular, it changes the electron and vibrational wave functions and the electrostatic energy of the adatom. These effects may also influence the transition probability and its dependence on the electrode potential. 

The nature of the intemuclear distance, r, is the object of interest in this chapter. In Eq. (5.1) it has the meaning of an instantaneous distance i.e., at the instant when a single electron is scattered by a particular molecule, r is the value that is evoked by the measurement in accordance with the probability density of the molecular state. Thus, when electrons are scattered by an ensemble of molecules in a given vibrational state v, characterized by the wave function r /v(r), the molecular intensities, Iv(s), are obtained by averaging the electron diffraction operator over the vibrational probability density. 

Figure 9.7 Vibrational energy levels determined from solution of the one-dimensional Schrodinger equation for some arbitrary variable 6 (some higher levels not shown). In addition to the energy levels (horizontal lines across the potential curve), the vibrational wave functions are shown for levels 0 and 3. Conventionally, the wave functions are plotted in units of (probability) with the same abscissa as the potential curve and an individual ordinate having its zero at the same height as the location of the vibrational level on the energy ordinate - those coordinate systems are explicitly represented here. Note that the absorption frequency typically measured by infrared spectroscopy is associated with the 0 —> 1 transition, as indicated on the plot. For the harmonic oscillator potential, all energy levels are separated by the same amount, but this is not necessarily the case for a more general potential...

It should be noted that (4.28) is only an approximation for the nuclear wave function. The perturbation terms (4.36) will mix into the nuclear wave function small contributions from harmonic-oscillator functions with quantum numbers other than v. These anharmonicity corrections to the vibrational wave function will add further to the probability of transitions with At) > 1. 

Actually, many other infrared transitions occur besides those allowed by the selection rule (6.74). The neglected terms in the expansion (6.66) will give transitions with a change of 2 or more in a given vibrational quantum number and transitions in which more than one vibrational quantum number changes moreover, anharmonicity corrections to the vibrational wave function will add to the probability of such transitions. Generally, the transitions (6.74) are the strongest. 

The M6X12 core (61 without terminal ligands) occurs in (3-Pd6Cl12235 and p-Pt6Cl12. These unique molecules are soluble in aromatic hydrocarbons, which probably function as electron donors to the metal atoms on the surface of the cluster.236 The same structure is proposed for the species [Bi OHJiJ6 on the basis of X-ray scattering from concentrated solutions, and vibrational spectra.237 The stoichiometry and the metal coordination stereochemistry are the same as those in the quite different cluster structure (6b). 

Figure 13.1 Vibrational levels and internuclear distance-probability functions for the ground state and first excited singlet of a diatomic molecule. Absorption and emission according to the Franck-Condon principle are illustrated. Adapted from N. J. Turro, Molecular Photochemistry, Addison-Wesley-—W. A. Benjamin, Reading, Mass., 1967. Reproduced by permission of Addison-Wesley,...

An internal conversion (IC) is observed when a molecule lying in the excited state relaxes to a lower excited state. This is a radiationless transition between two different electronic states of the same multiplicity and is possible when there is a good overlap of the vibrational wave functions (or probabilities) that are involved between the two states (beginning and final). 

Owing to the effects of mechanical anharmonicity - to which we shall refer in future simply as anharmonicity since we encounter electrical anharmonicity much less frequently -the vibrational wave functions are also modified compared with those of a harmonic oscillator. Figure 6.6 shows some wave functions and probability density functions (i//, i// )2 for an anharmonic oscillator. The asymmetry in ifjv and (i// i//,)2. compared with the harmonic oscillator wave functions in Figure 1.13, increases their magnitude on the shallow side of the potential curve compared with the steep side. 

Quantum mechanically there is a finite probability that inversion may occur even when the vibrational energy of the molecule is lower than the potential barrier Fmax. The vibrational wave functions for the parallel vibration [Eq. (1)] in the left (pl) and right (ips) potential minima penetrate the barrier and overlap to some extent. A given vibrational state is then described by a linear combination of and tpB into a symmetrical y>+ and an antisymmetrical yi function ... 

Figure 5.8. Stokes shift a) definition and b) dependence on the difference in equilibrium geometries of ground and excited states. Shown is the probability distribution in various vibrational levels, which is proportional to the square of the vibrational wave function (adapted from Philips and Salisbury, 1976).

An actual calculation of the S,- Sg jump probability requires quantum mechanical calculations of the time evolution of the wave packet representing the initial vibrational wave function as it passes through the funnel (Manthe and... 

A direct test of the vibrationally adiabatic approximation for H + H2 has also been made (Bowman et al.. 1973). This test was done by projecting accurate wavefunctions on the vibrationally adiabatic functions for zero curvature, and measuring deviations of the resulting probability weight from unity. The symmetric stretch motion was found to be adiabatic to within 10 % for total energies between 0-51 and 0-72 eV, but adiabaticity was lost at lower and higher energies. 

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